Definition:Rank (Linear Algebra)
Contents
Definition
Linear Transformation
Let $\phi$ be a linear transformation from one vector space to another.
Let the image of $\phi$ be finite-dimensional.
Then its dimension is called the rank of $\phi$ and is denoted $\map \rho \phi$.
Matrix
Definition 1
Let $K$ be a field.
Let $\mathbf A$ be an $m \times n$ matrix over $K$.
Then the rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the dimension of the subspace of $K^m$ generated by the columns of $\mathbf A$.
That is, it is the dimension of the column space of $\mathbf A$.
Definition 2
Let $K$ be a field.
Let $\mathbf A$ be an $m \times n$ matrix over $K$.
Let $\mathbf A$ be converted to echelon form $\mathbf B$.
Let $\mathbf B$ have exactly $k$ non-zero rows.
Then the rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is $k$.
Definition 3
Let $K$ be a field.
Let $\mathbf A$ be an $m \times n$ matrix over $K$.
The rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$ is the largest number of elements in a linearly independent set of rows of $\mathbf A$.
These definitions can be proved compatible (by viewing a matrix as a lin. transform., and maybe vice versa); such is a typical PW entry. Also, cf. Definition:Finite Rank Operator
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